Well, for the Monty Hall problem the argument that convinced me right away was this: Suppose there were 1000 doors and one prize. You chose one door, and Monty opens 998 other doors. In this situation, everybody would switch to the door Monty didn't open, although this setup is as symmetric as the 3-door case. From here it is much easier to convince people that the advantage of switching persists down
to 3 doors.

This argument follows the general strategy of arguing in
extremes. Another strategy (favored by physicists) is to simplify things
as much as possible (but not more!), sometimes even to the absurd.

The general question raised in "Monty Hall redux" is very interesting. Even in a world of
mathematical proofs and established experimental results, convincing people is an issue of its own. Sometimes even false arguments for a right statement are more convincing than all correct arguments.